Advertisement

From von Neumann to Wigner and beyond

  • J. S. Ben-BenjaminEmail author
  • L. Cohen
  • M. O. Scully
Regular Article
Part of the following topical collections:
  1. Non-equilibrium Dynamics: Quantum Systems and Foundations of Quantum Mechanics

Abstract

Historically, correspondence rules and quantum quasi-distributions were motivated by classical mechanics as a guide for obtaining quantum operators and quantum corrections to classical results. In this paper, we start with quantum mechanics and show how to derive the infinite number of quantum quasi-distributions and corresponding c-functions. An interesting aspect of our approach is that it shows how the c-numbers of position and momentum arise from the quantum operator.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.S. Ben-Benjamin, M.B. Kim, W.P. Schleich, W.B. Case, L. Cohen, Fortschr. Phys. 65, 1600092 (2017) CrossRefGoogle Scholar
  2. 2.
    M. Born, P. Jordan, Zeit. f. Phys. 34, 858 (1925) ADSCrossRefGoogle Scholar
  3. 3.
    H. Choi, W. Williams, IEEE Trans. ASSP 37, 862 (1989) CrossRefGoogle Scholar
  4. 4.
    T. Claasen, W. Mecklenbrauker, Phil. J. Res. 35, 372 (1980) Google Scholar
  5. 5.
    L. Cohen, J. Math. Phys. 7, 781 (1966) ADSCrossRefGoogle Scholar
  6. 6.
    L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, 1995) Google Scholar
  7. 7.
    L. Cohen, The Weyl Operator and its Generalization (Birkhauser, 2013) Google Scholar
  8. 8.
    L. Cohen, M. Scully, Found. Phys. 16, 295 (1986) ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Cohen, J. Mod. Optics 51, 2761 (2004) ADSCrossRefGoogle Scholar
  10. 10.
    R.J. Glauber, Phys. Rev. 131, 2766 (1963) ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    A.J.E.M. Janssen, Phil. J. Res. 37, 79 (1982) Google Scholar
  12. 12.
    J. Jeong, W. Williams, IEEE Trans. Sig. Process. 40, 402 (1992) ADSCrossRefGoogle Scholar
  13. 13.
    M. Kim, J.S. Ben-Benjamin, L. Cohen, J. Pseudo-Differ. Oper. Appl. 8, 661 (2017) MathSciNetCrossRefGoogle Scholar
  14. 14.
    J.G. Kirkwood, Phys. Rev. 45, 116 (1933) ADSCrossRefGoogle Scholar
  15. 15.
    H.W. Lee, Phys. Rep. 259, 147 (1995) ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    P. Loughlin, J. Pitton, L. Atlas, IEEE Trans. Sig. Process. 41, 750 (1993) ADSCrossRefGoogle Scholar
  17. 17.
    J.E. Moyal, Proc. Camb. Phil. Soc. 45, 99 (1949) ADSCrossRefGoogle Scholar
  18. 18.
    M.O. Scully, L. Cohen, in The Physics of Phase Space, edited by Y.S. Kim, W.W. Zachary (Springer Verlag, New York, 1987) Google Scholar
  19. 19.
    M. O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, 1997) Google Scholar
  20. 20.
    E.C.G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963) ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    H. Weyl, The Theory of Groups and Quantum Mechanics (E.P. Dutton and Co., 1928) Google Scholar
  22. 22.
    E.P. Wigner, Phys. Rev. 40, 749 (1932) ADSCrossRefGoogle Scholar
  23. 23.
    Y. Zhao, L. Atlas, R. Marks II, IEEE Trans. ASSP 38, 1084 (1990) CrossRefGoogle Scholar

Copyright information

© EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • J. S. Ben-Benjamin
    • 1
    Email author
  • L. Cohen
    • 2
    • 3
  • M. O. Scully
    • 1
    • 4
    • 5
  1. 1.Institute for Quantum Science and Engineering, Texas A&M UniversityCollege StationUSA
  2. 2.Hunter College, City University of New YorkNew YorkUSA
  3. 3.Graduate Center, City University of New YorkNew YorkUSA
  4. 4.Baylor UniversityWacoUSA
  5. 5.Princeton UniversityPrincetonUSA

Personalised recommendations